ONT Re: Differential Logic -- Series B

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DLOG.  Note B16

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I sometimes refer to the cactus lobe operators in the series
(), (x_1), (x_1, x_2), (x_1, x_2, x_3), ..., (x_1, ..., x_k)
as "boundary operators" and one of the reasons for this can
be seen most easily in the venn diagram for the k-argument
boundary operator (x_1, ..., x_k).  Figure 10 shows the
venn diagram for the 3-fold boundary form (x, y, z).

o-----------------------------------------------------------o
| U                                                         |
|                                                           |
|                      o-------------o                      |
|                     /               \                     |
|                    /                 \                    |
|                   /                   \                   |
|                  /                     \                  |
|                 /                       \                 |
|                o                         o                |
|                |            X            |                |
|                |                         |                |
|                |                         |                |
|                |                         |                |
|                |                         |                |
|             o--o----------o   o----------o--o             |
|            /    \%%%%%%%%%%\ /%%%%%%%%%%/    \            |
|           /      \%%%%%%%%%%o%%%%%%%%%%/      \           |
|          /        \%%%%%%%%/ \%%%%%%%%/        \          |
|         /          \%%%%%%/   \%%%%%%/          \         |
|        /            \%%%%/     \%%%%/            \        |
|       o              o--o-------o--o              o       |
|       |                 |%%%%%%%|                 |       |
|       |                 |%%%%%%%|                 |       |
|       |                 |%%%%%%%|                 |       |
|       |        Y        |%%%%%%%|        Z        |       |
|       |                 |%%%%%%%|                 |       |
|       o                 o%%%%%%%o                 o       |
|        \                 \%%%%%/                 /        |
|         \                 \%%%/                 /         |
|          \                 \%/                 /          |
|           \                 o                 /           |
|            \               / \               /            |
|             o-------------o   o-------------o             |
|                                                           |
|                                                           |
o-----------------------------------------------------------o
Figure 10.  Venn Diagram for (x, y, z)

In this picture, the "oval" (actually, octangular) regions that
are customarily said to be "indicated" by the basic propositions
x, y, z : B^3 -> B, that is, where the simple arguments x, y, z,
respectively, evaluate to true, are marked with the corresponding
capital letters X, Y, Z, respectively.  The proposition (x, y, z)
comes out true in the region that is shaded with per cent signs.
Invoking various idioms of general usage, one may refer to this
region as the indicated region, truth set, or fibre of truth
of the proposition in question.

It is useful to consider the truth set of the proposition (x, y, z)
in relation to the logical conjunction xyz of its arguments x, y, z.

In relation to the central cell indicated by the conjunction xyz,
the region indicated by "(x, y, z)" is composed of the "adjacent"
or the "bordering" cells.  Thus they are the cells that are just
across the boundary of the center cell, arrived at by taking all
of Leibniz's "minimal changes" from the given point of departure.

Jon Awbrey

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http://www.cs.bsu.edu/homepages/mighty/history.html
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